Algebra 1 Review- Key Concepts and Practice Problems
What You Actually Need to Know in Algebra 1
Most Algebra 1 courses dump too much information on you. Here's what actually matters—the concepts you'll use forever, not just until the final exam.
This guide cuts through the fluff. Master these basics and everything else gets easier.
Variables and Expressions
A variable is just a placeholder for a number you don't know yet. That's it. Nothing fancy.
Expressions combine numbers and variables with operations. Equations state that two expressions are equal.
Example:
3x + 7 is an expression
3x + 7 = 22 is an equation
The difference matters. Expressions don't have answers—you can't solve them. Equations you can solve.
Linear Equations
Linear equations graph as straight lines. Every Algebra 1 problem eventually comes back to these.
The Standard Form
Ax + By = C
But you'll mostly work with slope-intercept form:
y = mx + b
Where m is the slope and b is the y-intercept.
Solving One-Variable Equations
Get the variable alone. Whatever you do to one side, do to the other.
Example:
4x - 8 = 24
Add 8: 4x = 32
Divide by 4: x = 8
Check your work. Plug 8 back in: 4(8) - 8 = 32 - 8 = 24 ✓
Slope and Rate of Change
Slope tells you how steep a line is. It's the rate of change.
Slope formula:
m = (y₂ - y₁) / (x₂ - x₁)
Example: Points (2, 5) and (6, 13)
m = (13 - 5) / (6 - 2) = 8/4 = 2
Positive slope goes up left to right. Negative goes down. Zero is horizontal. Undefined is vertical.
Linear Inequalities
Same as equations, but with <, >, ≤, or ≥.
One critical difference: when you multiply or divide by a negative number, flip the inequality sign.
Example:
-3x > 12
Divide by -3: x < -4 (sign flips)
Graph inequalities on a number line. Closed circle for ≤ or ≥. Open circle for < or >.
Graphing Lines
You need three methods in your toolkit:
- Plot two points — find where x=0 and where y=0, connect
- Use slope-intercept — plot the y-intercept, then use slope to find another point
- X and Y intercepts — set x=0 to find y-intercept, set y=0 to find x-intercept
Practice all three. Tests will ask for any of them.
Systems of Equations
Two equations, two unknowns. You need both equations to be true at the same time.
Method 1: Substitution
Solve one equation for a variable, plug it into the other.
Example:
x + y = 10
y = 2x
Substitute: x + 2x = 10 → 3x = 10 → x = 10/3
y = 2(10/3) = 20/3
Method 2: Elimination
Add or subtract equations to cancel one variable.
Example:
2x + y = 12
x - y = 3
Add: 3x = 15 → x = 5
Plug back: 5 - y = 3 → y = 2
Both methods work. Pick whichever feels faster for the problem.
Exponents and Polynomials
Exponents are repeated multiplication. Know these rules cold:
- xᵃ · xᵇ = xᵃ⁺ᵇ
- xᵃ / xᵇ = xᵃ⁻ᵇ
- (xᵃ)ᵇ = xᵃˇᵇ
- x⁰ = 1 (anything to the zero power is 1)
- x⁻ᵃ = 1/xᵃ
Polynomials are sums of terms with variables raised to powers. Add and subtract by combining like terms—same variable, same exponent.
Multiply polynomials using FOIL for two binomials:
First, Outer, Inner, Last
Factoring
Factoring breaks down polynomials into products. It's the inverse of FOIL.
Factoring out GCF
Find the greatest common factor in every term and pull it out.
Example: 6x² + 9x
GCF is 3x
3x(2x + 3)
Factoring Quadratics
For x² + bx + c, find two numbers that multiply to c and add to b.
Example: x² + 5x + 6
What multiplies to 6 and adds to 5? 2 and 3.
(x + 2)(x + 3)
Quadratic Equations
These graph as parabolas—U-shaped curves.
The Quadratic Formula solves any quadratic equation ax² + bx + c = 0:
x = (-b ± √(b² - 4ac)) / 2a
Memorize this. You'll use it constantly.
The part under the square root (b² - 4ac) is the discriminant. It tells you:
- Positive = two real solutions
- Zero = one solution
- Negative = no real solutions
Solving Quadratics
You have four options. Use what's fastest:
- Factoring — fastest when it works cleanly
- Square roots — for equations like x² = 16
- Completing the square — useful for vertex form
- Quadratic formula — always works, always reliable
Practice Problems
1. Solve for x: 5(2x - 3) = 35
Answer: 5
2. Find the slope between (1, 4) and (5, 16):
Answer: 3
3. Factor: x² - 9
Answer: (x + 3)(x - 3)
4. Solve the system:
2x + y = 7
x - y = 2
Answer: x = 3, y = 1
5. Solve using quadratic formula: x² - 4x - 5 = 0
Answer: x = 5 or x = -1
Common Mistakes to Avoid
| Mistake | What to Do Instead |
|---|---|
| Dropping negative signs when distributing | Multiply every term, check your work |
| Forgetting to flip the inequality | Mark it in your work every time |
| Mixing up slope formula order | Subtract y's over x's consistently |
| Overusing the calculator | Practice basics until they're automatic |
| Skipping the check step | Plug your answer back in every time |
How to Actually Get Better
Reading this guide won't make you fluent. You need reps.
- Do 10+ problems daily — variety matters more than volume
- Time yourself — speed with accuracy is the goal
- Work without notes first — check after, not during
- Focus on weak areas — if factoring stalls you, drill factoring
- Use Khan Academy or Purplemath — free, decent explanations
Master the fundamentals and everything builds from there. This isn't optional groundwork—this is the math you'll use in Algebra 2, Precalc, and beyond.